Copula Calibration

TL;DR

This paper introduces copula calibration for multivariate probabilistic forecasts, providing methods to assess their calibration through the CopPIT and applying these to wind vector forecasts.

Contribution

It develops the concept of copula calibration and the CopPIT for multivariate forecast assessment, extending calibration tools to higher dimensions.

Findings

CopPIT histogram assesses multivariate forecast calibration.

Climatological copula calibration generalizes marginal calibration.

Application to wind forecasts demonstrates practical utility.

Abstract

We propose notions of calibration for probabilistic forecasts of general multivariate quantities. Probabilistic copula calibration is a natural analogue of probabilistic calibration in the univariate setting. It can be assessed empirically by checking for the uniformity of the copula probability integral transform (CopPIT), which is invariant under coordinate permutations and coordinatewise strictly monotone transformations of the predictive distribution and the outcome. The CopPIT histogram can be interpreted as a generalization and variant of the multivariate rank histogram, which has been used to check the calibration of ensemble forecasts. Climatological copula calibration is an analogue of marginal calibration in the univariate setting. Methods and tools are illustrated in a simulation study and applied to compare raw numerical model and statistically postprocessed ensemble forecasts of bivariate wind vectors.